Symmetry relationships

Given that mathematics is most idiomatic for describing symmetry
relationships, it is only too easy to get hung up on discussing
notions of mathematical determinism
and never get to the symmetries themselves. In the wake of the
previous article's remarks on reality
and indeterminate forms, it is natural to take up the subject of
symmetry again. One explicit technique by which some modern
composers have hoped to illuminate their deeper philosophical ideas
is through particular symmetries. In some cases, especially
retrograde melodic inversion, the symmetries can yield a static
quality which the composer presumably hoped to achieve. For
instance, composers such as Messiaen
use this technique extensively, to the point of constructing giant
symmetric building blocks for entire compositions. I want to begin
by discussing why this method is not very effective.

Although it is certainly possible to hold an entire piece of
music in one's mind, and regard it as a full entity as one would
a painting, that is not the inherent nature of the medium. Music
is laid out in time, and perceived
in time. Even if one feels comfortable regarding an entire piece
as described, it must first be perceived in time. This is as
distinct from the processes of the composer, who might conceive
aspects of the whole, and thus be tempted to try to recreate that
vision directly. However the process of perception interferes with
that idea, because one's memory is likewise time-dependent.
Therefore there is no direct way to relate such a unitary perception
(except visually, for ideas conducive to that expression), and
straightforward symmetry statements can actually serve to blunt
any underlying unity in the final perception through the distortion
of time-based apprehension. This is also true for prose. To put
it bluntly, the audience is bored by repetition. A simple inversion
is not sufficient variety, especially because it is laid out on
the same time scale. Progressive diminution is the most basic
means of restatement, because subsequent statements of equal length
will actually seem longer than the original. This fact is perhaps
expressed most clearly by the common role, whether consciously or
unconsciously, of "golden section" recapitulations. I
claim no special preeminence for that placement, but it is one
simple form of time relationship which works.

These remarks cannot be fully applied to smaller-scale elements.
For instance, the refrain in most song-forms is a direct repetition,
yet it is effective indeed. The refrain serves as a point of
departure, and remains effective by virtue of its relatively small
scale. This is in contrast to attempts to use similar repetition
schemes on larger elements. Indeed there are clear practical limits
of size to a refrain functioning as such, based essentially on the
idea of a "single statement" or a "point" in
the Renaissance sense, and similar lengths hold across cultures.
The length of the refrain was already taken to its limits in the
fifteenth century chanson, and with
it the development of those formal schemes. Such a simple factor
was as much responsible for the subsequent dramatic change in
art song writing as any other. Popular songs continue similar
forms and similar length constraints today. Even here, we frequently
see such "points" placed in different harmonic or rhythmic
guises, especially combinations via syncopation in the more advanced
settings. So the applicability of simple symmetry relationships
such as repetition is closely related to perceptual scale, and it
becomes less easy to indicate something on a larger scale by virtue
of such relationships. In short, any vision of
reality is necessarily mediated by
mundane time when transcribed from its indeterminate form, and
music at the highest level cannot hope to make light of this
constraint.

Symmetry breaking

Alongside the critique of large-scale symmetry, it is time to
turn to the smallest scale. Let us vaguely denote the harmonic
series as a symmetry relationship, such that a fully symmetric
sound would simply duplicate the entire series unaltered. Of course
real objects do not reproduce the full unaltered series, as they
can be not only truncated but altered in their frequency ratios.
This is basic physics, but the point is that this alteration from
a full series reflects a direct transformation of a vacuum into
the object itself. One can perhaps return to the topic of dualism
and indeterminate forms to realize how it is symmetry-breaking
which instantiates the physical world. Returning to music, even
the simplest piece (Scelsi aside)
makes use of more than a single sound and the corresponding series
of its sounding body. Subsequent sounds, forming to either
melody or harmony, serve to accentuate
certain pitches of this series, thus breaking the symmetry further.
In some cases, the geometric relationship between subsequent sounds
is so distant that symmetry is reduced essentially to a point.
This is the basis of some modern
theories, and of course reduction in symmetry opens a greater
variety of possibility. Even for most monophonic music, such as
Indian classical music, subsequent
notes relate in precise ways to previous ones, and specific symmetric
(or tetrachordal, if you will) relationships can be prescribed.

In all these cases, the main point is clear. Symmetry relationships
are so strongly linked to the act of
perception itself that they cannot be used as a straightforward
expression. They can work only as mediated by perception, and so
only in an imperfect form of a scope which straddles perceptual
divides. This is directly related to the fact that only a relatively
small range of stimulation densities
are apprehensible by ordinary humans, so it should come as no
surprise. Can symmetric relationships outside this scale have a
subconscious effect? It is a good question, and perhaps unanswerable.
Nonetheless, my inclination is to say no, the subconscious produces
affects only via repressed stimulation which passes through the
higher-level mind. More significantly, I believe that one must
also feel something directly in order to produce the sort of cascade
effect which actually alters thought
processes. In this era of excess stimulation, the last thing
one needs is to be immersed in something else which cannot be
apprehended. Such a thing brings only befuddlement, whereas the
difficult but perceivable brings the twin benefits of apprehension
and accomplishment.